{ localUrl: '../page/prime_element_ring_theory.html', arbitalUrl: 'https://arbital.com/p/prime_element_ring_theory', rawJsonUrl: '../raw/5m2.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'prime_element_ring_theory', edit: '3', editSummary: '', prevEdit: '2', currentEdit: '3', wasPublished: 'true', type: 'wiki', title: 'Prime element of a ring', clickbait: 'Despite the name, "prime" in ring theory refers not to elements which are "multiplicatively irreducible" but to those such that if they divide a product then they divide some term of the product.', textLength: '1211', alias: 'prime_element_ring_theory', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-08-21 07:33:25', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-07-27 20:21:56', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '41', text: '[summary: A prime element of a [3gq ring] is one such that, if it divides a product, then it divides (at least) one of the terms of the product.]\n\n[summary(Technical): Let $(R, +, \\times)$ be a [3gq ring] which is an [5md integral domain]. We say $p \\in R$ is *prime* if, whenever $p \\mid ab$, it is the case that either $p \\mid a$ or $p \\mid b$ (or both).]\n\nAn element of an [-5md] is *prime* if it has the property that $p \\mid ab$ implies $p \\mid a$ or $p \\mid b$.\nEquivalently, if its generated [ideal_ring_theory ideal] is [prime_ideal prime] in the sense that $ab \\in \\langle p \\rangle$ implies either $a$ or $b$ is in $\\langle p \\rangle$.\n\nBe aware that "prime" in ring theory does not correspond exactly to "[4mf prime]" in number theory (the correct abstraction of which is [5m1 irreducibility]). \nIt is the case that they are the same concept in the ring $\\mathbb{Z}$ of [48l integers] ([5mf proof]), but this is a nontrivial property that turns out to be equivalent to the [-5rh] ([alternative_condition_for_ufd proof]).\n\n# Examples\n\n# Properties\n\n- Primes are always [5m1 irreducible]; a proof of this fact appears on the [5m1 page on irreducibility], along with counterexamples to the converse.\n- ', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'PatrickStevens', 'EricBruylant' ], childIds: [], parentIds: [ 'algebraic_ring' ], commentIds: [], questionIds: [], tagIds: [ 'stub_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19057', pageId: 'prime_element_ring_theory', userId: 'PatrickStevens', edit: '3', type: 'newEdit', createdAt: '2016-08-21 07:33:25', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17925', pageId: 'prime_element_ring_theory', userId: 'EricBruylant', edit: '2', type: 'newEdit', createdAt: '2016-08-01 22:12:38', auxPageId: '', oldSettingsValue: '', newSettingsValue: 'linebreak to fix summaries' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17601', pageId: 'prime_element_ring_theory', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-07-27 20:21:57', auxPageId: 'algebraic_ring', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17602', pageId: 'prime_element_ring_theory', userId: 'PatrickStevens', edit: '0', type: 'newTag', createdAt: '2016-07-27 20:21:57', auxPageId: 'stub_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17599', pageId: 'prime_element_ring_theory', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-07-27 20:21:56', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }